Unique manifold structure on a group

Question

Is it the case that a group is compatible with at most one manifold structure. That is, any two manifolds the make the group into a topological group are homeomorphic?

I know the reverse claim is false in the sense that many different Lie groups can have the same topology. But also that it true for compact simply connected manifolds, in the sense that they carry at most one group structure, see my question: Unique group structure on compact connected manifold

Answer 1

The question has already been asked and answered here:

Does a Lie group's group structure (not Lie group structure) determine its topology?

The answer is basically yes it is unique unless you do some weird axiom of choice stuff (and for compact case it is always unique).